Available Research Projects

Harmonic analysis originated with the study of Fourier series and Fourier transforms, but now also includes the study of wavelets, with many applications in signal and image processing. Interesting questions connect harmonic analysis to analysis on metric spaces, fractals, geometric measure theory, graph theory, group theory, number theory and partial differential equations.

Duration: Summer research or 1 semester honours project
Summer projects may be funded by an award

Algebra and Number Theory, like: Galois Theory;  p-adic numbers; elliptic curves.

Project 1: Extracting system-level information from data
How do we understand the dynamics of a system if we only have data from it? Can we predict beyond just what we see in the data? This stream incorporates machine learning and dynamical systems analysis to learn differential equations from data, extrapolate beyond the training set, identify important features of the system, and control the output. 

Project 2: Pattern formation on random networks
Collective behaviour over networks of interacting agents is ubiquitous. The question is: can we predict patterns of synchronization when we can only describe the network structure probabilistically? This project brings together graph theory, probability and statistics, and dynamical systems to understand how patterns form on random networks. 

Project 1: Determinants in infinite dimensions: Fredholm determinants.
Reference: "Trace Ideals and their applications", by Barry Simon.
Prerequisites: linear algebra and functional analysis (464)
Duration: 1 semester (scalable)

Project 2: “Orthogonal polynomials and applications”,
Reference: Orthogonal polynomials and random matrices: a Riemann-Hilbert approach, Deift, P. A.
Prerequisite: all 3xx level done
Duration: 1 semester

Project 3: “Fuchsian groups and hyperbolic geometry”
Reference: (for example) The geometry of Discrete Groups (A. F. Beardon)
Duration: 1 semester

Assessment of climate change risk on natural catastrophes. Measuring and quantifying risk and uncertainty rising from climate change on insurable extreme risks.

Dr. Brugiapaglia offers to (co-)supervise honours projects in the areas of data science and computational mathematics. Projects can be theory-oriented (e.g., involving the theoretical analysis of state-of-the-art algorithms), computationally-oriented (e.g., the implementation of new computational methods and numerical tests on synthetic or real-world data), or a mix of both. Specific areas of interest include deep learning, compressed sensing, numerical approximation methods, and their applications.

Duration: 1 semester